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Theorem ldualgrplem 29880
Description: Lemma for ldualgrp 29881. (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
ldualgrp.d  |-  D  =  (LDual `  W )
ldualgrp.w  |-  ( ph  ->  W  e.  LMod )
ldualgrp.v  |-  V  =  ( Base `  W
)
ldualgrp.p  |-  .+  =  o F ( +g  `  W
)
ldualgrp.f  |-  F  =  (LFnl `  W )
ldualgrp.r  |-  R  =  (Scalar `  W )
ldualgrp.k  |-  K  =  ( Base `  R
)
ldualgrp.t  |-  .X.  =  ( .r `  R )
ldualgrp.o  |-  O  =  (oppr
`  R )
ldualgrp.s  |-  .x.  =  ( .s `  D )
Assertion
Ref Expression
ldualgrplem  |-  ( ph  ->  D  e.  Grp )

Proof of Theorem ldualgrplem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ldualgrp.f . . . 4  |-  F  =  (LFnl `  W )
2 ldualgrp.d . . . 4  |-  D  =  (LDual `  W )
3 eqid 2435 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 ldualgrp.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 29861 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2440 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2436 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 eqid 2435 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
943ad2ant1 978 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  W  e.  LMod )
10 simp2 958 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  x  e.  F )
11 simp3 959 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  y  e.  F )
121, 2, 8, 9, 10, 11ldualvaddcl 29865 . 2  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  ( x
( +g  `  D ) y )  e.  F
)
13 ldualgrp.r . . . . 5  |-  R  =  (Scalar `  W )
14 eqid 2435 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
154adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
16 simpr2 964 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
17 simpr3 965 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
181, 13, 14, 2, 8, 15, 16, 17ldualvadd 29864 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  =  ( y  o F ( +g  `  R
) z ) )
1918oveq2d 6089 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
20 simpr1 963 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  F )
211, 2, 8, 15, 16, 17ldualvaddcl 29865 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  e.  F )
221, 13, 14, 2, 8, 15, 20, 21ldualvadd 29864 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) ( y ( +g  `  D
) z ) )  =  ( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) ) )
231, 2, 8, 15, 20, 16ldualvaddcl 29865 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  e.  F )
241, 13, 14, 2, 8, 15, 23, 17ldualvadd 29864 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( ( x ( +g  `  D
) y )  o F ( +g  `  R
) z ) )
251, 13, 14, 2, 8, 15, 20, 16ldualvadd 29864 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  =  ( x  o F ( +g  `  R
) y ) )
2625oveq1d 6088 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y )  o F ( +g  `  R
) z )  =  ( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z ) )
2713, 14, 1, 15, 20, 16, 17lfladdass 29808 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2824, 26, 273eqtrd 2471 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2919, 22, 283eqtr4rd 2478 . 2  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x ( +g  `  D ) ( y ( +g  `  D ) z ) ) )
30 eqid 2435 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
31 ldualgrp.v . . . 4  |-  V  =  ( Base `  W
)
3213, 30, 31, 1lfl0f 29804 . . 3  |-  ( W  e.  LMod  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
334, 32syl 16 . 2  |-  ( ph  ->  ( V  X.  {
( 0g `  R
) } )  e.  F )
344adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
3533adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
36 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
371, 13, 14, 2, 8, 34, 35, 36ldualvadd 29864 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  ( ( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x ) )
3831, 13, 14, 30, 1, 34, 36lfladd0l 29809 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x )  =  x )
3937, 38eqtrd 2467 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  x )
40 eqid 2435 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
41 eqid 2435 . . 3  |-  ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  =  ( z  e.  V  |->  ( ( inv g `  R ) `
 ( x `  z ) ) )
4231, 13, 40, 41, 1, 34, 36lflnegcl 29810 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  e.  F
)
431, 13, 14, 2, 8, 34, 42, 36ldualvadd 29864 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  o F ( +g  `  R ) x ) )
4431, 13, 40, 41, 1, 34, 36, 14, 30lflnegl 29811 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  o F ( +g  `  R
) x )  =  ( V  X.  {
( 0g `  R
) } ) )
4543, 44eqtrd 2467 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( V  X.  { ( 0g `  R ) } ) )
466, 7, 12, 29, 33, 39, 42, 45isgrpd 14822 1  |-  ( ph  ->  D  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3806    e. cmpt 4258    X. cxp 4868   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   Grpcgrp 14677   inv gcminusg 14678  opprcoppr 15719   LModclmod 15942  LFnlclfn 29792  LDualcld 29858
This theorem is referenced by:  ldualgrp  29881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lfl 29793  df-ldual 29859
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